The sum `n` terms of the series `1/(sqrt(1)+sqrt(3))+1/(sqrt(3)+sqrt(5))+1/(sqrt(5)+sqrt(7))+` is
`sqrt(2n+1)` (b) `1/2sqrt(2n+1)` (c) `1/2sqrt(2n+1)-1` (d) `1/2{sqrt(2n+1)-1}`

Explanation

Given,`1/(sqrt(1)+sqrt(3))+1/(sqrt(3)+sqrt(5))+1/(sqrt(5)+sqrt(7))+.....1/(sqrt(2n-1)+sqrt(2n+1))`

=`1/(sqrt(3)+sqrt(1))+1/(sqrt(5)+sqrt(3))+1/(sqrt(7)+sqrt(5))+.....1/(sqrt(2n+1)+sqrt(2n-1))`

=`(sqrt(3)-sqrt(1)]/(sqrt(3)+sqrt(1).(sqrt(3)-sqrt(1)))+1(sqrt(5)-sqrt(3)]/[(sqrt(5)+sqrt(3)).(sqrt(5)-sqrt(3))]+.....[1.(sqrt(2n+1)-sqrt(2n-1))]/[(sqrt(2n-1)+sqrt(2n+1)).(sqrt(2n+1)-sqrt(2n-1))]`

=`(sqrt(3)-sqrt(1))/(sqrt(3)^2-sqrt(1)^2)+(sqrt(5)-sqrt(3))/(sqrt(5)^2-sqrt(3)^2)+.....+(sqrt(2n+1)-sqrt(2n-1))/((sqrt2n+1)^2-sqrt((2n-1))^2)`

=`(sqrt(3)-sqrt(1))/2+(sqrt(5)-sqrt(3))/(2)+....+(sqrt(2n+1)-sqrt(2n-1))/2`

=`1/2(sqrt(3)-sqrt(1)+sqrt(5)-sqrt(3)+....sqrt(2n+1)-sqrt(2n-1))`

=`1/2(sqrt(2n+1)-1)`